{
  "id": "2207.01385",
  "version": "v1",
  "published": "2022-07-04T13:08:54.000Z",
  "updated": "2022-07-04T13:08:54.000Z",
  "title": "Fractional Bloom boundedness and compactness of commutators",
  "authors": [
    "Tuomas Hytönen",
    "Tuomas Oikari",
    "Jaakko Sinko"
  ],
  "comment": "26 pages",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Let $T$ be a non-degenerate Calder\\'on-Zygmund operator and let $b:\\mathbb{R}^d\\to\\mathbb{C}$ be locally integrable. Let $1<p\\leq q<\\infty$ and let $\\mu^p\\in A_p$ and $\\lambda^q\\in A_q,$ where $A_{p}$ denotes the usual class of Muckenhoupt weights. We show that \\begin{align*} \\|[b,T]\\|_{L^p_{\\mu}\\to L^q_{\\lambda}}\\sim \\|b\\|_{\\operatorname{BMO}_{\\nu}^{\\alpha}},\\qquad [b,T]\\in \\mathcal{K}(L^p_{\\mu}, L^q_{\\lambda})\\quad\\mbox{iff}\\quad b\\in \\operatorname{VMO}_{\\nu}^{\\alpha}, \\end{align*} where $L^p_\\mu=L^p(\\mu^p)$ and $\\alpha/d = 1/p-1/q,$ , the symbol $\\mathcal{K}$ stands for the class of compact operators between the given spaces, and the fractional weighted $\\operatorname{BMO}_{\\nu}^{\\alpha}$ and $\\operatorname{VMO}_{\\nu}^{\\alpha}$ spaces are defined through the following fractional oscillation and Bloom weight \\begin{align*} \\mathcal{O}_{\\nu}^{\\alpha}(b;Q) = \\nu^{-\\alpha/d}(Q)\\Big(\\frac{1}{\\nu(Q)}\\int_Q |b-\\langle b\\rangle_Q|\\Big),\\qquad \\nu = \\big(\\frac{\\mu}{\\lambda}\\big)^{\\beta},\\quad \\beta = (1+\\alpha/d)^{-1}. \\end{align*} The key novelty is dealing with the off-diagonal range $p<q$, whereas the case $p=q$ was previously studied by Lacey and Li. However, another novelty in both cases is that our approach allows complex-valued functions $b$, while other arguments based on the median of $b$ on a set are inherently real-valued.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-04T13:08:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional bloom boundedness",
      "compactness",
      "commutators",
      "non-degenerate calderon-zygmund operator",
      "muckenhoupt weights"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}